Box interaction strength approximation

We construct the molecular product states which have the spatial wavefunctions 

The dipole-dipole interaction V has a diagonal term, which slightly shifts the energies, and an off diagonal term, which induces the collisional transitions. Since Vdepends on r, Vis time dependent, and the wavefunction is given by the solution to the time dependent Schroedinger equation 

When the two possible states are those given by Eq. (14.13) the general form of the solution to Eq. (14.14) is 

In Eqs. (14.16) the matrix elements of V are time dependent, since the intemuclear distance is a function of time, as are the coefficients CA(t) and CB(f). In general, Eqs. (14.16) cannot be solved analytically, but in several special cases analytic solutions can be obtained. 

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